Group #47979349 — subgroups

Subgroups — #1960×C₂₇₇ #47979349

Order 35456.

308 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Q₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#103×C₂₇₇	C₂	1	4	yes	maximal
3	C₂×#97×C₂₇₇	C₂	1	1	yes	maximal, characteristic
4	#382×C₂₇₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₇₇	C₂²	1	2	yes	—
6	C₂²×C₈×C₂₇₇	C₂²	1	2	yes	—
7	C₄⋊Q₈×C₂₇₇	—	2	2	no	—
8	C₄⋊Q₈×C₂₇₇	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₂₇₇	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
12	#103×C₂₇₇	—	2	8	no	—
13	#97×C₂₇₇	—	2	2	no	—
14	#95×C₂₇₇	—	2	2	no	—
15	C₂×Q₈×C₂₇₇	—	2	2	no	—
16	C₂×Q₈×C₂₇₇	—	2	4	no	—
17	C₂×Q₈×C₂₇₇	—	2	4	no	—
18	C₂×Q₈×C₂₇₇	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₂₇₇	—	2	2	no	—
20	C₂²×C₄×C₂₇₇	—	2	2	no	—
21	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
22	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₂₇₇	—	2	2	no	—
24	C₂×C₈×C₂₇₇	—	2	2	no	—
25	C₂×C₈×C₂₇₇	D₄	1	4	yes	—
26	C₄⋊C₄×C₂₇₇	—	2	8	no	—
27	C₄⋊C₄×C₂₇₇	—	2	4	no	—
28	C₄²×C₂₇₇	C₂×C₄	1	2	yes	—
29	C₄²×C₂₇₇	D₄	1	2	yes	—
30	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₂₇₇	—	2	8	no	—
32	Q₈×C₂₇₇	—	4	4	no	—
33	C₂×C₄×C₂₇₇	—	2	4	no	—
34	C₂×C₄×C₂₇₇	—	2	4	no	—
35	C₂×C₄×C₂₇₇	—	4	2	no	—
36	C₂×C₄×C₂₇₇	—	4	2	no	—
37	C₂×C₄×C₂₇₇	—	4	2	no	—
38	C₂×C₄×C₂₇₇	—	4	4	no	—
39	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	4	yes	—
40	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₇₇	#41	1	2	yes	—
44	C₈×C₂₇₇	—	2	4	no	—
45	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₇₇	#124	1	1	yes	characteristic
47	C₂²×C₂₇₇	#127	1	1	yes	characteristic
48	C₂²×C₂₇₇	#137	1	4	yes	—
49	C₄×C₂₇₇	—	2	4	no	—
50	C₄×C₂₇₇	—	4	4	no	—
51	C₄×C₂₇₇	—	8	2	no	—
52	C₄×C₂₇₇	C₄×D₄	1	2	yes	—
53	C₄×C₂₇₇	#121	1	2	yes	—
54	C₂×C₂₇₇	#390	1	1	yes	characteristic
55	C₂×C₂₇₇	#419	1	2	yes	—
56	C₂×C₂₇₇	#491	1	2	yes	—
57	C₂×C₂₇₇	#497	1	2	yes	—
58	C₂₇₇	#1960	1	1	yes	characteristic, direct product factor, Sylow
59	#1960	C₂₇₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×C₄⋊Q₈	C₂×C₂₇₇	1	1	yes	characteristic
61	C₂×#103	C₂×C₂₇₇	1	4	yes	—
62	C₂×#97	C₂×C₂₇₇	1	1	yes	characteristic
63	#382	C₂×C₂₇₇	1	1	yes	characteristic
64	C₂²×Q₈	C₂²×C₂₇₇	1	2	yes	—
65	C₂²×C₈	C₂²×C₂₇₇	1	2	yes	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	C₄×C₂₇₇	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	C₂²×C₂₇₇	1	2	yes	—
70	C₂×C₄²	C₂²×C₂₇₇	1	1	yes	characteristic
71	#103	—	2	8	no	—
72	#97	—	2	2	no	—
73	#95	—	2	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×C₄×C₂₇₇	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	D₄×C₂₇₇	1	2	yes	—
81	C₂²×C₄	C₂³×C₂₇₇	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	D₄×C₂₇₇	1	4	yes	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	2	4	no	—
87	C₄²	C₂×C₄×C₂₇₇	1	2	yes	—
88	C₄²	D₄×C₂₇₇	1	2	yes	—
89	C₂³	C₂×D₄×C₂₇₇	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	C₂²⋊C₄×C₂₇₇	1	4	yes	—
99	C₂×C₄	C₂²×C₄×C₂₇₇	1	1	yes	characteristic
100	C₂×C₄	C₂×D₄×C₂₇₇	1	2	yes	—
101	C₂×C₄	C₂×D₄×C₂₇₇	1	1	yes	characteristic
102	C₂×C₄	#41×C₂₇₇	1	2	yes	—
103	C₈	—	2	4	no	—
104	C₂²	C₂×C₂²⋊C₄×C₂₇₇	1	1	yes	characteristic
105	C₂²	#124×C₂₇₇	1	1	yes	characteristic
106	C₂²	#127×C₂₇₇	1	1	yes	characteristic
107	C₂²	#137×C₂₇₇	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	8	2	no	—
111	C₄	C₄×D₄×C₂₇₇	1	2	yes	—
112	C₄	#121×C₂₇₇	1	2	yes	—
113	C₂	#390×C₂₇₇	1	1	yes	characteristic
114	C₂	#419×C₂₇₇	1	2	yes	—
115	C₂	#491×C₂₇₇	1	2	yes	—
116	C₂	#497×C₂₇₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₂₇₇

C₂

yes

maximal, characteristic

C₂×#103×C₂₇₇

C₂

yes

maximal

C₂×#97×C₂₇₇

C₂

yes

maximal, characteristic

#382×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₇₇

C₂²

yes

—

C₂²×C₈×C₂₇₇

C₂²

yes

—

C₄⋊Q₈×C₂₇₇

—

C₄⋊Q₈×C₂₇₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₇₇

—

C₂×C₄⋊C₄×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

#103×C₂₇₇

—

#97×C₂₇₇

—

#95×C₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

C₂×C₄

yes

—

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

D₄

yes

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

C₂×C₄

yes

—

C₄²×C₂₇₇

D₄

yes

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

Q₈×C₂₇₇

—

Q₈×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₄×D₄

yes

—

C₄×C₂₇₇

#121

yes

—

C₂×C₂₇₇

#390

yes

characteristic

C₂×C₂₇₇

#419

yes

—

C₂×C₂₇₇

#491

yes

—

C₂×C₂₇₇

#497

yes

—

C₂₇₇

#1960

yes

characteristic, direct product factor, Sylow

#1960

C₂₇₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₂₇₇

yes

characteristic

C₂×#103

C₂×C₂₇₇

yes

—

C₂×#97

C₂×C₂₇₇

yes

characteristic

#382

C₂×C₂₇₇

yes

characteristic

C₂²×Q₈

C₂²×C₂₇₇

yes

—

C₂²×C₈

C₂²×C₂₇₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₄×C₂₇₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₇₇

yes

—

C₂×C₄²

C₂²×C₂₇₇

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₂₇₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₇₇

yes

—

C₂²×C₄

C₂³×C₂₇₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₂₇₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×C₂₇₇

yes

—

C₄²

D₄×C₂₇₇

yes

—

C₂³

C₂×D₄×C₂₇₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₂₇₇

yes

—

C₂×C₄

C₂²×C₄×C₂₇₇

yes

characteristic

100

C₂×C₄

C₂×D₄×C₂₇₇

yes

—

101

C₂×C₄

C₂×D₄×C₂₇₇

yes

characteristic

102

C₂×C₄

#41×C₂₇₇

yes

—

103

C₈

—

104

C₂²

C₂×C₂²⋊C₄×C₂₇₇

yes

characteristic

105

C₂²

#124×C₂₇₇

yes

characteristic

106

C₂²

#127×C₂₇₇

yes

characteristic

107

C₂²

#137×C₂₇₇

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

C₄×D₄×C₂₇₇

yes

—

112

C₄

#121×C₂₇₇

yes

—

113

C₂

#390×C₂₇₇

yes

characteristic

114

C₂

#419×C₂₇₇

yes

—

115

C₂

#491×C₂₇₇

yes

—

116

C₂

#497×C₂₇₇

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₇₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₇₇
Frattini, cosocle	C₂²×C₄	C₂³×C₂₇₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₇₇ ◁ C₂×C₄²×C₂₇₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₇₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₂₇₇

Frattini, cosocle

C₂²×C₄

C₂³×C₂₇₇

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₇₇ ◁ C₂×C₄²×C₂₇₇ ◁ G

N	H ≅ G/N	type
8. C₄⋊Q₈×C₂₇₇	C₄	semidirect
58. C₂₇₇	#1960	direct (N × H)
59. #1960	C₂₇₇	direct (N × H)
67. C₄⋊Q₈	C₄×C₂₇₇	semidirect

H ≅ G/N

type

8. C₄⋊Q₈×C₂₇₇

C₄

semidirect

58. C₂₇₇

#1960

direct (N × H)

59. #1960

C₂₇₇

direct (N × H)

67. C₄⋊Q₈

C₄×C₂₇₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊Q₈×C₂₇₇	2
C₂×#103×C₂₇₇	2
C₂×#97×C₂₇₇	2
#382×C₂₇₇	2
#1960	277

subgroup

index

C₂×C₄⋊Q₈×C₂₇₇

C₂×#103×C₂₇₇

C₂×#97×C₂₇₇

#382×C₂₇₇

#1960

277

Subgroups — #1960×C₂₇₇ #47979349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1960×C277 #47979349

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1960×C₂₇₇ #47979349